Finding Rightmost Eigenvalues of Large, Sparse, Nonsymmetric Parameterized Eigenvalue Problems

Transcription

1 Finding Rightmost Eigenvalues of Large, Sparse, Nonsymmetric Parameterized Eigenvalue Problems AMSC Final Report Minghao Wu AMSC Program Dr. Howard Elman Department of Computer Science 1

2 Problem Statement To find the rightmost eigenvalues of: Ax = λbx. where matrices A and B are Real N-by-N Large Sparse Nonsymmetric Depend on one or several parameters

3 Application To determine the stability of the linearized system of the form [1] : B x& = Ax The steady state solution x* is - stable, if all the eigenvalues of Ax = λbx have negative real parts; - unstable, otherwise. 3

4 Basic Arnoldi Algorithm and Implicitly Restarted Arnoldi Properties of basic Arnoldi algorithm Matrix transformation Implicitly restarted Arnoldi 4

5 Arnoldi algorithm [1] Eigensolver - iterative method -based on Krylov subspace k - dimensional Krylov subspace: K k (A,u 1 ) = span{u Au 1 1 A u 1 A k u 1 } - residual of computed eigenpairs is orthogonal to K k - only solves standard eigenvalue problem Ax = λx - converges to well-separated extremal eigenvalues 5

6 Matrix Transformation Shift Invert Transformation [1] T SI = (A-σB) -1 B Ax = λbx λ T SI x= θx θ = 1/(λ-σ) 6

7 Computational Result Example: Olmstead model [] ut = S xx + cu xx + Ru u bst = (1 c) u S u = S = 0 at x = 0, π. u: velocity, S: a quantity related to viscoelastic forces b, c, R: parameters. R: Rayleigh number b =, c = 0.1, R = 0.6, N = 1000, k = 0, σ = 0: rightmost eigenvalues: λ 1, = 0 ± 0.447i residual: Ax i λ i x i / x i = , i = 1, 3 7

8 Implicitly Restarted Arnoldi Motivation - large Krylov subspace is not practical - singular B gives rise to spurious eigenvalues Basic idea of IRA Filter out unwanted eigendirections from the starting vector of Arnoldi algorithm by applying shifted QR algorithm to a small upper Hessenberg matrix [3] 8

9 Computational Result Example [4] : K C T C x 0 M = λ 0 0 x 0 - K (00 00), C (00 100) and M (00 00) are of full rank - eigenvalues lie between -3 and 50 - mimics the eigenproblem arises from Navier-Stokes equation Exact Eigenvalues i.11 ± 1.156i.5036 ± 0.064i i.1318 ± 0.356i.1081 ± 1.353i k = 10, σ = 60 Computed Eigenvalues (IRA) i.11 ± 1.156i.5036 ± 0.064i i.1318 ± 0.356i.1081 ± 1.353i Computed Eigenvalues (basic Arnoldi) ± i (residual 10-4 ) i i i ±.707i ± 0.753i

10 Arnoldi Algorithm with Iterative Linear System Solver How to solve (A - σb)w = b efficiently? (A σb: large, sparse, nonsymmetric) 10

11 GMRES Method GMRES stands for the Generalized Minimal RESidual method. Based on Krylov subspace Solves the following minimization problem min x x 0 + K m b Ax where x 0 is the initial guess and K m is the m dimensional Krylov subspace 11

12 Goal Preconditioning of Ax = b cluster the eigenvalues of the original matrix Basic idea solve M -1 Ax = M -1 b instead of Ax = b where 1) M approximates A ) M -1 is easy to apply Example: incomplete LU factorization (ILU) - Basic idea: drop certain entries in the complete LU factors of the matrix, eg., entries < some threshold - ILU(0): select the allowed fill to exactly match the sparsity pattern of the original matrix A 1

13 Computational Result Example: Olmstead model, N = 5000, R = 4.7, λ 1 = 4.510, σ = 5 solver: GMRES with ILU(0) Method No relaxation Relaxation with BF Number of GMRES iteration at every outer iteration Bouras and Fraysse s relaxation strategy [6] : tolerance GMRES = 10 - ε / r k-1 Run time (sec.) Relative error e e

14 Application in The Study of Dynamical Equilibrium The detection of: multiple steady states change of stability Hopf bifurcation phenomena 14

15 15 Tubular Reactor Model Tubular reactor model [7] with boundary condition ( ) ( ) ( ) ( ) 0,1 / exp 1 / exp = = s on BDy s s Pe t Dy s y s y Pe t y h m θ γ γ θ θ β θ θ θ θ γ γ ( ) ( ) , = = = = = = s at s s y s at Pe s y Pe s y h m θ θ θ y: velocity; θ: temperature; Pe m, Pe h, B, D, γ, β: parameters. D: Damkohler number

16 Computation of Solution Path [8] Published Result [7] Computed Result Pe m = Pe h = 5, B = 0.5, γ = 5, β = 5 16

17 Eigenvalue Problem From Navier-Stokes equations D driven-cavity problem Reynolds number and stability Detection of eigenvalues with large imaginary part 17

18 D Driven-Cavity Problem Equations [5] : u t υ r u r r + u u + p r u = = 0 0 ( ) u x u y u r =, u x = 1 x 4 at y = 1. : velocity, p: pressure, υ>0: kinematic viscosity. Reynolds number: UL R = υ A quantitative measure of the relative contributions of viscous diffusion and convection. 18

19 Reynolds Number And Stability Reynolds Number Rightmost Eigenvalues ± 1.6i ± 1.3i ± 1.37i ± 1.3i ± 1.47i Rightmost eigenvalues change from real to complex at R around As R increases, steady state solution loses its stability. Hopf bifurcation happens at R around Imaginary parts are much larger than real parts (in modulus). Q -Q 1 macroelement, h = -4, nonlinear residual<

20 Conclusions What do we have - Arnoldi and Implicitly Restarted Arnoldi code - Arnoldi with iterative linear system solver - Codes for discretizing several commonly used benchmark problems - Continuation code for single-parameter nonlinear dynamical systems - Validated results for various computational tasks performed on these benchmark problems What s next - Detection of eigenvalues with large imaginary parts - Preconditioning of Arnoldi applied to Navier-Stokes equations - Look at a Navier-Stokes equation with low critical Reynolds number 0

21 Acknowledgement The Implicitly Restarted Arnoldi code is written by Fei Xue, a PhD student of Department of Mathematics, University of Maryland. The software we used to discretize the Navier-Stokes equation and compute its steady state solution is IFISS, Incompressible Flow Iterative Solution Software, developed by: Howard Elman Department of Computer Science, University of Maryland David Silvester Department of Mathematics, University of Manchester Andy Wathern Oxford University, Computing Laboratory 1

22 Reference [1] Meerbergen, K & Roose, D 16 Matrix transformation for computing rightmost eigenvalues of large sparse non-symmetric eigenvalue problems. SIAM J. Numer. Anal. 16, [] Olmstead, W. E., Davis, W. E., Rosenblat, S. H., & Kath, W. L. 186 Bifurcation with memory. SIAM J.Appl. Math. 40, [3] Stewart, G. W. 001 Matrix algorithms, volume II: eigensystems. SIAM. [4] Meergergen, K & Spence, A 17 Implicitly restarted Arnoldi with purification for the shift-invert transformation. Math. Comput. 66, [5] Elman, H & Silvester, D & Wathen, A 005 Finite elements and fast iterative solvers with applications in incompressible fluid dynamics. Oxford University Press [6] Simoncini, V 005 Variable accuracy of matrix-vector products in projection methods for eigencomputation. SIAM J. Numer. Anal. 43, [7] Heinemann, R & Poore, A 181 Multiplicity, stability, and oscilatory dynamics of the tubular reactor.chemical Engineering Science, [8] Spence, A & Graham, Ivan G., Numerical Methods for Bifurcation Problems.